Influence of Transient Processes in the Stator Winding and Parameters on the Static Stability of Synchronous Reluctance Machines
Abstract Generated abstract
This paper examines how design and operating parameters affect the static stability region of synchronous reluctance machines, with particular attention to stator winding transients and short-circuited rotor windings. Using a previously developed stability method and numerical boundary calculations, it analyzes the roles of resistance, the ratio of direct to quadrature reactance, air gap, inertia, rotor winding parameters, and load angle. The results indicate that overload capacity is governed by steady-state parameters, while self-oscillation depends on both steady and transient conditions, with active resistance and rotor winding configuration playing major roles. Neglecting stator winding transients simplifies the characteristic equation and may be acceptable in some cases, but the study finds that such transients can substantially enlarge the predicted self-oscillation region.
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Influence of Transient Processes in the Stator Winding and Parameters on the Static Stability of Synchronous Reluctance Machines
E. V. Kononenko
(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)
In order to design a synchronous reluctance machine (SRM) that will operate stably in a specified mode, it is necessary to know the influence of design parameters on the region of static stability of these machines. The influence of SRM parameters on the region of stable operation was investigated by the method set forth in [1]. The boundaries of the static-stability region were calculated on the “Minsk-1” electronic digital computer.
The investigations showed that the influence of individual parameters on the size of the region of stable operation is manifested as follows.
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The overload capacity (slip) of an SRM depends only on the parameters of the established operating mode. As the ratio \(\frac{x_d}{x_q}\) increases, the overload capacity increases. An increase in the active resistance in the stator-winding circuit leads to a decrease in overload capacity.
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The size of the self-oscillation region of an SRM is determined by the parameters of both the established and the transient modes of operation.
Self-oscillation is caused by the presence of active resistance in the stator-winding circuit. An increase in active resistance narrows the region of stable operation of an SRM.
With an increase in the ratio \(\frac{x_d}{x_q}\), the self-oscillation region expands. Figure 1 gives the stability boundary \(\frac{x_d}{x_q}=f(r)\), calculated under the condition that the remaining SRM parameters are constant \((H=100\) electrical sec, \(\Theta_0=0^\circ;\ T_d=40\) electrical sec, \(T_q=10\) electrical sec, \(x_q=0.45;\ x_q'=0.2;\ x_q''=0.2)\).
With a decrease in the synchronous inductive reactance \(x_d\), the region of stable operation narrows. The reactance \(x_d\) is inversely proportional to the magnitude of the air gap. Consequently, as the air gap increases, the self-oscillation region expands. Figure 1 gives the stability boundary \(x_d=f(r)\), calculated under the condition that \(\frac{x_d}{x_q}=5.0\), while the remaining parameters are the same as in the preceding case.
An increase in the moment of inertia of the rotating masses expands the region of stable operation of an SRM. Figure 2 gives the boundary of stable operation when the mechanical constant of the rotating masses is varied,
Fig. 1. Boundaries of the stable-operation region constructed for variation of the resistance \(x_d\) and the ratio \(\frac{x_d}{x_q}\).
Fig. 2. Construction of the boundary of the stable-operation region for variation of the mechanical constant \(H\).
calculated for an SRM having the same parameters as in the preceding cases, and \(x_d=2.33\); \(x_q=0.45\). Short-circuited rotor windings have a substantial influence on the region of stable operation [1, Fig. 1]. In the absence of short-circuited windings on the rotor, the SRM cannot operate stably. By an appropriate choice of the parameters of the rotor winding, the occurrence of self-oscillation can be made practically impossible.
The size of the self-oscillation region also depends on the load of the machine. In the investigation of static stability, the load value in the equations is determined by the angle \(\Theta_0\) [1]. In this case, the regions of stable and unstable operation of the SRM are constructed in the plane of the parameters \(\Theta_0\) and \(r\). The boundary separating the generator and motor modes of operation of the SRM in these coordinates is determined by the points corresponding to ideal no-load operation. The value of the angle \(\Theta_0\) for the no-load mode is determined from the equality
\[ \tg \theta_0=-\frac{r}{x_d}. \tag{1} \]
When investigating the influence of load on the region of stable operation of an SRM, it is expedient to construct the stability boundaries as a function of the angle between the axis of the rotor pole and the axis of the magnetomotive force of the stator winding, \(\beta_0\). For known values of the parameters and \(\Theta_0\), this angle is determined from the relation
\[ \tg \beta_0=\frac{x_d\sin\theta_0+r\cos\theta_0}{x_q\cos\theta_0-r\sin\theta_0}. \tag{2} \]
In Fig. 3 the boundaries of stable operation are constructed in the plane of the parameters \(\beta_0\) and \(r\), calculated for an SRM having the parameters: \(H=200\) el. sec; \(x_d=2.33\); \(x_q=0.45\); \(x'_d=x''_q=0.2\). From the curves shown it follows that, as the load increases, the possibility of the occurrence of self-oscillation decreases. If the rotor has a winding only along the longitudinal axis, stable operation at small loads is impossib-
na. The presence of a complete winding on the rotor causes the self-oscillation region to shift toward larger values of the active resistance.
Fig. 3. Boundaries of the region of stable operation, plotted as a function of the load angle \(\beta_0\)
\(1\) — \(T_d = 40\) electrical sec, \(T_q = 0\);
\(2\) — \(T_d = 40\) electrical sec, \(T_q = 10\) electrical sec;
\(3\) — slip boundary;
— — — without allowance for transient processes in the stator winding.
In studying the static stability of a synchronous reluctance machine, it is of interest to consider the influence of transient processes in the stator winding. For small disturbances, it may be assumed that the transformer emfs and the slip emfs have practically no effect on the conditions for self-oscillation and may be neglected [2].
Then the linearized system of equations for a synchronous reluctance machine having a complete winding on the rotor will have the form [1]:
\[ U \cos \theta_0 \Delta \theta + r \Delta i_d - x_q(p)\Delta i_q = 0; \]
\[ U \sin \theta_0 \Delta \theta + x_d(p)\Delta i_d + r \Delta i_q = 0; \tag{3} \]
\[ Hp^2 \Delta \theta + [x_d(p)-x_q]i_{q0}\Delta i_{d0} + [x_d-x_q(p)]i_{d0}\Delta i_q = 0. \]
Here the quantities with the subscript zero characterize the initial steady-state operating condition; quantities with the sign \(\Delta\) are due to small oscillations of the rotor.
Forming and equating to zero the determinant of the system of equations (3), after the corresponding transformations the characteristic equation can be represented in the form
\[ a_0p^4 + a_1p^3 + a_2p^2 + a_3p + a_4 = 0, \tag{4} \]
\[ a_0 = H(r^2 + x_d''x_q'')T_dT_q; \]
\[ a_1 = H\left[r^2(T_d+T_q) + x_d''x_qT_d + x_dx_q''T_q\right]; \]
\[ \begin{gathered} a_2=H\left(x_d x_q+r^2\right)+B\left(x_d''\cos\theta_0-r\sin\theta_0\right)T_d U i_{d0}+\\ {}+A\left(x_q''\sin\theta_0+r\cos\theta_0\right)T_q U i_{q0}; \end{gathered} \]
\[ \begin{gathered} a_3=\left[Bx_d+(x_d-x_q)x_d''T_d\right]U\cos\theta_0\, i_{d0}+\\ {}+\left[Ax_q-(x_d-x_q)x_q''T_q\right]U\sin\theta_0\, i_{q0} +\left[A-(x_d-x_q)T_q\right]rU\cos\theta_0\, i_{q0}-\\ {}-\left[B+(x_d-x_q)T_d\right]rU\sin\theta_0\, i_{d0}; \end{gathered} \]
\[ a_4=(x_d-x_q)U\left[(x_d\cos\theta_0-r\sin\theta_0)i_{d0}-(x_q\sin\theta_0+r\cos\theta_0)i_{q0}\right]; \]
\[ A=(x_q-x_d'')T_d;\qquad B=(x_d-x_q'')T_q; \]
\[ i_{q0}=\frac{Ux_q\cos\theta_0-Ur\sin\theta_0}{x_d x_q+r^2}; \qquad i_{d0}=\frac{Ux_d\sin\theta_0+Ur\cos\theta_0}{x_d x_q+r^2}. \]
Comparing (4) with the corresponding equation in [1], we see that, when transient processes in the stator winding are neglected, the degree of the characteristic equation is reduced by two. The expressions for the coefficients of the characteristic equation are simplified. All this considerably facilitates calculations and the analysis of the stability of an SRM.
The boundaries of the slip region do not depend on transient processes in the stator winding, since the expression for the coefficient \(\alpha_6\) does not change.
As the investigations have shown, when transient processes in the stator winding are neglected, the boundaries of self-oscillation are mainly determined by the parameters of the steady operating mode of the SRM. Thus, in the case where the rotor has a short-circuited winding only along the longitudinal axis, the Hurwitz determinant for equation (4) can be represented in the form
\[ \Delta_r=\frac{H\left(x_d-x_d''\right)T_d U^2}{x_d x_q+r^2} \left\{r^4\cos^2\theta_0+r^3x_d\sin2\theta_0+\right. \]
\[ \left. {}+r^2x_q\left[(2x_d-x_q)\sin^2\theta_0-(x_d-2x_q)\cos^2\theta_0\right]+ \right. \]
\[ \left. {}+rx_q^3\sin2\theta_0+x_d\cdot x_q^3\sin^2\theta_0 \right\} \tag{5} \]
From this equation it is evident that, for a very small value of the mechanical constant \(H\), or in the absence of short-circuited windings on the rotor, stable operation of the SRM is impossible. For finite values of \(H\), \(X_d''\), \(T_d\), the boundaries of stable operation are determined by the conditions under which the expression in braces in equation (5) is equal to zero. Consequently, in the case under consideration, the boundaries of stable operation are determined only by the parameters of the steady operating mode of the SRM.
As the investigations have shown, transient processes in the stator winding in a number of cases considerably expand the region of self-oscillation. When the rotor has a winding only along the longitudinal axis, the change in the boundary of stable operation may be insignificant, as is seen from Fig. 3.
REFERENCES
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E. V. Kononenko. Investigation of the static stability of synchronous reluctance machines. In the present collection.
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N. I. Sokolov et al. Application of analog computing machines in power systems, Energiya Publishing House, 1964.